Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

CMAP); Jaime San (CMM); Pierre Collet (CPHT); Sylvie M\'el\'eard (MERGE

REVIEW 2 major objections 2 minor 1 cited by

This paper proves that, under spectral hypotheses on the Feynman-Kac semigroup, measure-valued real-trait birth-death processes have a well-defined long-time behavior in all three regimes, including a Yaglom limit and Q-process for the proc

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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For real-trait birth-death processes, the paper proves moment recurrences, extinction-probability asymptotics, and convergence to a Yaglom limit and Q-process when the population is conditioned to survive.

T0 review reviewed 2026-08-06 challenge →

load-bearing objection Plausible and meaningful extension of measure-valued birth-death asymptotics to jump-diffusive traits, but the central Yaglom claim is gated on unstated Feynman-Kac hypotheses; the critical regime needs scrutiny. the 2 major comments →

arxiv 2508.04089 v1 pith:GEMZFZD6 submitted 2025-08-06 math.PR

Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

Pierre Collet (CPHT) , Sylvie M\'el\'eard (MERGE , CMAP) , Jaime San (CMM) This is my paper

classification math.PR MSC 60J8060J8560F05

keywords measure-valued birth-death processYaglom limitQ-processFeynman-Kac semigroupextinction probabilitycriticality regimestrait-structured populationlong-time asymptotics

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish the full long-time asymptotic picture for a class of continuous-time birth-death processes in which each individual carries a real-valued trait that evolves as a one-dimensional Markov process. The authors prove a new recurrence for the factorial moments and the extinction probability, and obtain their exponential asymptotics in the subcritical, critical, and supercritical regimes. Their main result is that the process conditioned on non-extinction converges in law to a Q-process, giving a Yaglom limit in this infinite-dimensional setting. The value is a tractable description of the distribution of a surviving population with continuous trait variation, including the quasi-stationary distribution that appears before eventual extinction.

Core claim

Under suitable hypotheses on the Feynman-Kac semigroup, the paper shows that the measure-valued birth-death process has a complete long-time behavior that is uniform across all three criticality regimes. The recurrence expresses the factorial moments of the population measure and the extinction probability through integrals of the semigroup against product functions, and the leading exponential rates are governed by the principal eigenvalue of that semigroup. Conditioning the process on non-extinction, the authors construct a Q-process, the process biased by the ground-state eigenfunction, and prove that the law of the population given survival up to time $t$ converges to a Yaglom limit in t

What carries the argument

The Feynman-Kac semigroup of the linearized branching generator. This is the semigroup obtained by weighting the one-individual trait dynamics by the multiplicative functional of the birth and death rates. Its principal eigenvalue is the Malthusian parameter separating subcritical, critical, and supercritical regimes, and exponential convergence to its ground state is what forces the moment recurrences and the conditioning limit. The Q-process is defined by conditioning via the ground-state eigenfunction.

Load-bearing premise

The entire long-time picture rests on the paper's 'suitable hypotheses' on the Feynman-Kac semigroup—in particular, that it has a distinguished principal eigenvalue and a spectral gap; if the trait dynamics do not converge exponentially to a ground state, the Yaglom limit need not exist.

What would settle it

Choose a subcritical birth-death process whose Feynman-Kac semigroup has no spectral gap (for instance, a mutation kernel with a power-law tail that destroys exponential mixing). If the law of the population measure conditioned on survival to time t fails to converge, or converges to a measure different from the Q-process invariant measure, the claimed Yaglom limit is false for that process.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

The Q-process gives a pathwise description of the population conditional on indefinite survival, so in the subcritical regime the population that has not yet died out follows a Markov chain with an explicit generator.
The moment recurrence supplies exact formulas for factorial moments and extinction probabilities, which in applied work can be used to estimate trait-dependent branching rates from census data.
The three example classes show that the assumptions are satisfied by natural mechanisms, including diffusion and jump mutations, so the Yaglom limit is not an empty statement.
In the supercritical regime, conditioning on non-extinction leads to a Q-process describing the limiting immortal population, which can be used to study the genealogy of the trait distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The spectral-gap hypothesis could likely be weakened to a mixing condition on the trait process, covering heavy-tailed mutation kernels; the paper does not address this relaxation.
The Q-process constructed here may provide the ancestral lineage distribution of the population, analogous to ancestral graphs in population genetics, which would make the results useful for inference of selection.
A numerical test would be to simulate a subcritical process with a given trait-dependent branching rate, compute the empirical distribution of population size given survival up to time t, and check convergence to the predicted Yaglom distribution; deviations would indicate where the spectral hypothesis fails.
The recurrence might extend to age-structured or multi-dimensional trait birth-death processes, where the one-dimensional Markov assumption is the main algebraic obstacle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Plausible and meaningful extension of measure-valued birth-death asymptotics to jump-diffusive traits, but the central Yaglom claim is gated on unstated Feynman-Kac hypotheses; the critical regime needs scrutiny.

read the letter

At a glance: the paper is a credible extension of the measure-valued birth-death asymptotic program to traits driven by jumps as well as diffusion. The contributions it claims—a new moment recurrence, time asymptotics, convergence in law of the conditioned process, and a Q-process/Yaglom limit—are exactly the kind of results that matter for structured population models. The inclusion of jumps is genuinely new relative to the diffusive-trait literature, and the three example classes are a good way to show the hypotheses are not empty.

The soft spot is the phrase "under suitable hypotheses on the Feynman-Kac semigroup." That is not a defect by itself—every theorem has assumptions—but here it is load-bearing. The whole asymptotic program depends on those hypotheses, and the abstract doesn't state them. The critical regime is where I'd point the referee. In many critical birth-death and branching models, the relevant semigroup decays algebraically, not exponentially, so a spectral gap is absent and the Yaglom limit typically requires a different normalization. If the paper's hypotheses allow for polynomial decay, then the claimed "Yaglom limit" may not be the classic one, and the abstract's phrasing would be misleading. We can't tell from the abstract whether the critical example actually satisfies the conditions. That's the main thing I'd want the referee to check.

I also can't audit novelty from an abstract. The "new recurrence" may or may not overlap with earlier results in the purely diffusive case; that's a citation check, not a red flag.

Who is this for: people who work on measure-valued branching processes, structured populations, and quasi-stationary limits. If correct, it's a useful tool and I'd cite it. It's not a paradigm shift, but it's a solid piece of the established research program.

My recommendation: don't desk-reject. Send it to an expert referee, and make sure the referee is told to verify the spectral hypotheses, the critical regime, and the examples. My own verdict is open because I've only seen the abstract, but the problem is well-posed and the authors are doing honest work.

Referee Report

2 major / 2 minor

Summary. The paper studies a class of continuous-time measure-valued birth-and-death processes with a one-dimensional trait that evolves as a Markov process with jumps between birth/death events. The abstract announces results in the critical, subcritical, and supercritical regimes: a new recurrence for moments and extinction probability, time asymptotics, and convergence in law of the process conditioned on non-extinction, yielding an infinite-dimensional Yaglom limit and Q-process. These claims are made 'under suitable hypotheses on the Feynman-Kac semigroup.' Three classes of examples are said to satisfy the hypotheses.

Significance. The extension of Yaglom-limit theory to measure-valued, trait-structured processes is a natural and challenging problem with applications in mathematical biology and stochastic population dynamics. If the results are correct, this would constitute a substantial contribution, unifying behavior across all three regimes in an infinite-dimensional setting. The paper promises new recurrence relations and asymptotic laws that go beyond the scalar birth-death case. However, the significance is contingent on the validity and scope of the unstated Feynman-Kac hypotheses, especially in the critical regime, where spectral behavior may differ from the subcritical and supercritical cases.

major comments (2)

[Abstract (suitable hypotheses)] The paper's central theorems are conditional on unspecified hypotheses on the Feynman-Kac semigroup. This is a load-bearing premise: the Yaglom limit and Q-process existence typically require a principal eigenvalue and appropriate spectral convergence for the associated linear semigroup. The abstract does not state these conditions, nor does it indicate how they are verified in the three example classes. In many critical birth-death and branching models, the Feynman-Kac semigroup exhibits algebraic rather than exponential decay, so either the hypotheses must be relaxed (possibly changing the normalization of the limit) or it must be shown that the critical example still satisfies them. The full text must state the hypotheses explicitly and verify them for each example, in particular the critical one, before the theorem's claims can be accepted.
[Convergence in law in infinite dimension] The claim of convergence in law for the conditioned measure-valued process requires a tightness argument on the space of measures. The abstract does not specify the topology on the measure space or the method used to establish tightness. Without this technical component, the Yaglom-limit statement is incomplete. The full text must provide the precise space, metric, and proof of convergence in law, especially because the process is measure-valued and infinite-dimensional.

minor comments (2)

[Abstract clarity] The abstract would benefit from a one-sentence characterization of the three regimes (critical, subcritical, supercritical) in terms of a model parameter, such as the Malthusian growth rate or a comparable spectral quantity, to orient the reader.
[Terminology] The phrase 'new recurrence' would be more informative if it indicated what aspect is new relative to existing scalar or measure-valued results, e.g., in terms of moment order or trait-dependence.

Circularity Check

0 steps flagged

No circularity evident from the abstract; the results are conditional on stated hypotheses, not on the conclusions.

full rationale

The available evidence is the abstract only; the full derivation chain is not accessible. The abstract states theorems under 'suitable hypotheses on the Feynman-Kac semigroup', which is a conditional mathematical statement rather than a circular one: the hypotheses are premises, and the claimed conclusions (moment recurrences, asymptotics, Yaglom limit, Q-process) are intended to be derived from them. No equation or construction is shown in which an output equals an input by definition, and no fitted parameter is renamed as a prediction. The reader's caution about unstated spectral hypotheses concerns correctness or applicability, not circularity: a conditional theorem does not become circular merely because its hypotheses are not fully enumerated in the abstract. No self-citation is visible at this level, and there is no quoted text exhibiting a reduction of the claimed result to its own assumptions. Therefore, under the rule that circularity must be demonstrated with quoted evidence and specific reductions, the honest finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No free parameters or invented entities appear in the abstract; this is a pure mathematics paper with no data fitting. The main hidden cost is the domain assumption on the Feynman-Kac semigroup, which is load-bearing and not specified in the abstract. Everything else rests on standard semigroup and martingale theory.

axioms (3)

domain assumption Trait processes between birth/death events are one-dimensional Markov processes including diffusion and jumps.
Defines the model class in the abstract; all theorems apply only within this class.
domain assumption The Feynman-Kac semigroup satisfies 'suitable hypotheses' (unspecified), presumably a spectral-gap or principal-eigenvalue condition with exponential convergence to a ground state.
The asymptotic results, including the Yaglom limit and Q-process existence, are explicitly conditional on these hypotheses, which the abstract does not state.
standard math Standard stochastic analysis background, including Feynman-Kac formalism, martingale characterizations of measure-valued processes, and spectral theory of one-dimensional semigroups, is used without proof.
Usual unstated background for this genre of probability paper.

reviewed 2026-08-06 · how reviews work

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Cite this review

Pith. "Pith review of Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes." pith.science (2026). https://pith.science/paper/GEMZFZD6

@misc{pith2026250804089,
  author       = {Pith},
  title        = {Pith review of: Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GEMZFZD6}},
  note         = {Machine review of arXiv:2508.04089}
}

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read the original abstract

In this article we study the long time behaviour of measure-valued birth and death processes in continuous time, where the dynamics between jumps are one-dimensional Markov processes including diffusion and jumps. We consider the three regimes, critical, subcritical and supercritical. Under suitable hypotheses on the Feynman-Kac semigroup, we prove a new recurrence for the moments and the extinction probability, their time asymptotics and the convergence in law for the measure-valued birth and death process conditioned to non extinction, leading to the existence of Q-process and Yaglom limit (in this infinite dimensional setting). We develop three classes of natural examples where our results apply.

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Long-Time Behaviors of Branching-Diffusion Processes via Spectral Analysis
math.PR 2026-04 unverdicted novelty 7.0

Establishes exponential convergence rates for the total mass of branching-diffusion processes and characterizes their quasi-stationary distributions via a novel spectral transformation and heat kernel estimates.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages · cited by 1 Pith paper

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M. Bertoldi & L. Lorenzi . Analytical Methods for Markov Semigroups. Chapman & Hall/CRC, 2007

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Pith/arXiv arXiv 2024
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D. Dawson, B. Maisonneuve & J. Spencer . \'Ecole d'\'Et\'e de P robabilit\'es de S aint- F lour XXI ---1991 , Lecture Notes in Mathematics, vol. 1541, Springer-Verlag, 1991

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S.C. Harris, E. Horton, A.E. Kyprianou, M. Wang . Yaglom limit for critical nonlocal branching Markov processes. The Annals of Probab. 50 (6), (2022), 2373--2408

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Harris, E

S.C. Harris, E. Horton, A.E. Kyprianou. Stochastic Methods for the Neutron Transport Equation II: Almost sure growth. Annals of Applied Probability (2020), Vol. 30, No. 6, 2573-2612

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B. Henry, S. M\'el\'eard & V.C. Tran . Time reversal of spinal processes for linear and non-linear branching processes near stationarity, Electron. J. Probab. 28, (2023), Paper no. 32, 27, DOI 10.1214/23-ejp911

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Horton, A.E

E. Horton, A.E. Kyprianou, D. Villemonais . Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics. Ann. Appl. Probab. (2020) 30 , No. 6, 2815--2845

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S. M \'e l \'e ard & V.C. Tran . Nonlinear historical superprocess approximations for population models with past dependence, Electron. J. Probab. 17 (2012), no. 47, 32, DOI 10.1214/EJP.v17-2093

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This paper was first reviewed by deepseek-v4-flash on August 6, 2026.

[1] [1]

Arendt, A

W. Arendt, A. Grabosch, G. Greiner, U. Moustakas, R. Nagel, U. Schlotterbeck, U. Groh, H. Lotz, F. Neubrander . One Parameter Semigroups of Positive Operators. Lecture notes in Mathematics 1184. Springer 1986. See also Arendt W., Gl\"uck J. Positive irreducible semigroups and their long-time behaviour. Phil. Trans. R. Soc. A (2020), 378: 20190611

work page 1986

[2] [2]

K. B. Athreya, P. E. Ney . T.E. Harris and branching processes, Ann. Probab. 39(2), (2011), 429--434

work page 2011

[3] [3]

Perturbations of Positive Semigroups with Applications

J.Banasiak, L.Arlotti . Perturbations of Positive Semigroups with Applications. Springer 2006

work page 2006

[4] [4]

Bansaye, J.F

V. Bansaye, J.F. Delmas, L. Marsalle & V.C. Tran . Limit theorems for markov processes indexed by continuous time Galton-Watson trees, Ann. Appl. Probab. 21 (6), (2011), p. 2263--2314

work page 2011

[5] [5]

Bansaye, A

V. Bansaye, A. Lambert . New approaches of source-sinks metapopulations decoupling the roles of demography and dispersal. Theoretical Population Biology 88, (2013), 31--46

work page 2013

[6] [6]

Bertoldi & L

M. Bertoldi & L. Lorenzi . Analytical Methods for Markov Semigroups. Chapman & Hall/CRC, 2007

work page 2007

[7] [7]

Billingsley

P. Billingsley. Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 1979, Wiley

work page 1979

[8] [8]

Calvez, B

V. Calvez, B. Henry, S. M\'el\'eard & V.C. Tran . Dynamics of lineages in adaptation to a gradual environmental change, Ann. H. Lebesgue 5 (2022), p. 729--777

work page 2022

[9] [9]

Champagnat & S

N. Champagnat & S. M \'e l \'e ard . Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol. 55 (2007), no. 2, p. 147--188

work page 2007

[10] [10]

B. Cloez B. , Limit theorems for some branching measure-valued processes, Adv. in Appl. Probab. 49 (2), (2017), p. 549--580

work page 2017

[11] [11]

B. Cloez B. & P. Gabriel P. . On an irreducibility type condition for the ergodicity of nonconservative semigroups, C. R. Math. Acad. Sci. Paris 358 (2020), no. 6, p. 733--742

work page 2020

[12] [12]

Collet, S

P. Collet, S. M\'el\'eard & J. San Mart\'in . Branching diffusion processes and spectral properties of Feynman-Kac semigroup, arXiv:2404.09568 [math.PR] (2024)

Pith/arXiv arXiv 2024

[13] [13]

Dawson, B

D. Dawson, B. Maisonneuve & J. Spencer . \'Ecole d'\'Et\'e de P robabilit\'es de S aint- F lour XXI ---1991 , Lecture Notes in Mathematics, vol. 1541, Springer-Verlag, 1991

work page 1991

[14] [14]

Dawson & Perkins E.A

D. Dawson & Perkins E.A. . Historical processes, Mem. Amer. Math. Soc. vol. 93, 1991

work page 1991

[15] [15]

Fournier and S

N. Fournier and S. M\'el\'eard . A microscopic probabilistic description of a locally regulated population and macroscopic approximations, The Annals of Applied Probability 14 (2004), no. 4, 1880--1919

work page 2004

[16] [16]

Gonzalez, E

I. Gonzalez, E. Horton, A.E. Kyprianou . Asymptotic moments of spatial branching processes. Probability Theory and Related Fields (2022), 184 no. 3-4, 805--858

work page 2022

[17] [17]

Hardy, S.C

R. Hardy, S.C. Harris . A spine approach to branching diffusions with applications to L ^p -convergence of martingales, S \'e minaire de probabilit \'e s XLII, Lecture Notes in Math. 1979, Springer (2009), p. 281--330

work page 1979

[18] [18]

Harris, E

S.C. Harris, E. Horton, A.E. Kyprianou, M. Wang . Yaglom limit for critical nonlocal branching Markov processes. The Annals of Probab. 50 (6), (2022), 2373--2408

work page 2022

[19] [19]

Harris, E

S.C. Harris, E. Horton, A.E. Kyprianou. Stochastic Methods for the Neutron Transport Equation II: Almost sure growth. Annals of Applied Probability (2020), Vol. 30, No. 6, 2573-2612

work page 2020

[20] [20]

Henry, S

B. Henry, S. M\'el\'eard & V.C. Tran . Time reversal of spinal processes for linear and non-linear branching processes near stationarity, Electron. J. Probab. 28, (2023), Paper no. 32, 27, DOI 10.1214/23-ejp911

work page doi:10.1214/23-ejp911 2023

[21] [21]

Horton, A.E

E. Horton, A.E. Kyprianou, D. Villemonais . Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics. Ann. Appl. Probab. (2020) 30 , No. 6, 2815--2845

work page 2020

[22] [22]

T. Kato . Perturbation Theory of Linear Operators. Springer, 1966

work page 1966

[23] [23]

Kurtz, R

T. Kurtz, R. Lyons, R. Pemantle & Y. Peres . A conceptual proof of the K esten- S tigum theorem for multi-type branching processes, Classical and modern branching processes, IMA Vol. Math. Appl. vol. 84, Springer (1997), p. 181--185

work page 1997

[24] [24]

Z. Li . Measure-Valued Branching Markov Processes. Springer 2011

work page 2011

[25] [25]

Lyons, R

R. Lyons, R. Pemantle & Y. Peres . Conceptual proofs of L L criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), no. 3, p. 1125--1138

work page 1995

[26] [26]

A. Marguet . Uniform sampling in a structured branching population, Bernoulli 25 (2019), no. 4A, p. 2649--2695

work page 2019

[27] [27]

M \'e l \'e ard & V.C

S. M \'e l \'e ard & V.C. Tran . Nonlinear historical superprocess approximations for population models with past dependence, Electron. J. Probab. 17 (2012), no. 47, 32, DOI 10.1214/EJP.v17-2093

work page doi:10.1214/ejp.v17-2093 2012

[28] [28]

A. Pazy . Semigroups of Linear Operatorand Applications to Partial Differential Equations. Springer 1983

work page 1983

[29] [29]

u rich, 1994), Birkh\

E. Perkins . Measure-valued diffusions and interactions, Proceedings of the I nternational C ongress of M athematicians V ol. 1, 2 ( Z \" u rich, 1994), Birkh\" a user, Basel 1995, p. 1036--1046

work page 1994

[30] [30]

Eulerian Numbers, Birkh\"auser, 2015

K.Petersen . Eulerian Numbers, Birkh\"auser, 2015

work page 2015

[31] [31]

Introduction to Probability Models, Academic Press, 2007

S.Ross . Introduction to Probability Models, Academic Press, 2007

work page 2007

[32] [32]

Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, vol

M.Reed, B.Simon . Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, vol. 2, Academic Press, 1975

work page 1975

[33] [33]

H.L. Royden . Real analysis. Third edition Macmillan Publishing Company, New York, 1988

work page 1988

Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it